To set the stage, in the Section  we give a short review of the essential RG technique, which generalises the usual approach to the exact (aka the non-perturbative functional) RG (see [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]} for reviews) for the effective average action (EAA) by allowing for field to be reparameterised along the RG flow. In Section  we revisit S. Weinberg's formulation of asymptotic safety which emphasises the manner in which essential couplings enter expressions for observables.
In Section  we derive the generalised flow equation for quantum gravity that takes into account the freedom to reparameterise the quantum metric along the RG flow.
Indeed, the flow equation will contain a new ingredient: the RG kernel. This quantity encodes the description about how the fields are reparameterised along the flow.
Then, we write down a systematic derivative expansion of the diffeomorphism invariant part of the EAA and the covariant RG kernel. In particular, we expand the EAA to fourth order in derivatives and the RG kernel to second order.
In Section , we analyse the GFP properties: in particular, from this analysis we determine that the vacuum energy is inessential. The advantage of studying the GFP consists of the fact that it is a free fixed point and the results can be obtained without relying on approximations.
After having found the inessential couplings at the GFP, in Section  we discuss the properties of the universality class that contains the GFP and all the trajectories that have the same essential couplings. In such a subspace of the theory space, the propagator evaluated on conformally flat spacetime possesses the same form as the one at the GFP.
Up to order four in the derivative expansion (apart from the topological Gauss–Bonnet term) only Newton's constant is essential in this universality class.
In particular, any fixed point on these trajectories has the degrees of freedom of the GFP.
In Section , we study the RG flow of QEG in the minimal essential scheme at orders two and four of the derivative expansion. Our investigation confirms the existence of the Reuter fixed point: this implies that higher derivative terms coming from the operators \(\sqrt{\det g}\, R^2\)  and \(\sqrt{\det g}\, R_{\mu \nu }R^{\mu \nu }\)  are inessential in the universality class of the GFP. Moreover, this means that, contrary to the expectations based on perturbative renormalisability, the existence of the Reuter fixed point in the minimal essential scheme suggests that a possible UV-completion of the gravitational theory does not require additional degrees of freedom.
In Section  we draw our conclusions and discuss the outlook for future investigations of quantum gravity using the essential RG. The derivation of RG equations in the minimal essential scheme for QEG at fourth order in the derivative expansion for a generic dimension and a generic regulator cut-off are presented in Appendix REF .
